Generalized Ulam–Hyers–Rassias Stability Results of Solution for Nonlinear Fractional Differential Problem with Boundary Conditions

A Naimi,B Tellab,Y Altayeb,A Moumen,Adel Ouannas

doi:10.1155/2021/7150739

Abstract

The problem of existence and generalized Ulam–Hyers–Rassias stability results for fractional differential equation with boundary conditions on unbounded interval is considered. Based on Schauder’s fixed point theorem, the existence and generalized Ulam–Hyers–Rassias stability results are proved, and then some examples are given to illustrate our main results.

Highlights

Introduction and Position of Problem ere are various, not equivalent, definitions of fractional derivatives according to Grunwald Letnikov, Weil, Caputo, and Riemann–Liouville, etc
Fractional differential equations, which are often encountered in mathematical modeling of various processes in natural and technical sciences, play an important role in describing many phenomena in physics, bioengineering, and engineering applications. e properties of such equations were investigated in many reviews
Shen et al [9] considered the existence of solution for boundary value problem of nonlinear multipoint fractional differential equation:

Summary

Literature Overview

Fractional differential equations, which are often encountered in mathematical modeling of various processes in natural and technical sciences, play an important role in describing many phenomena in physics, bioengineering, and engineering applications. e properties of such equations were investigated in many reviews (among them, we refer [1,2,3,4,5,6]). Shen et al [9] considered the existence of solution for boundary value problem of nonlinear multipoint fractional differential equation:. SM Ulam in 1940 was the first to raise the question of stability for functional equations. After his lecture, this question became popular for many specialists in mathematical analysis. As far as we know, most authors discussed Ulam stability of some fractional differential problem on bounded/unbounded intervals, while the present paper discusses the existence of solutions and stability in the sense of Ulam–Hyers–Rassias for nonlinear fractional differential equations boundary conditions, for which research is just beginning, please see [18,19,20,21,22,23]

Preliminaries

Existence Result

Stability Result